In a recent paper, the last three authors showed that a game-theoretic p-harmonic function v is characterized by an asymptotic mean value property with respect to a kind of mean value $$\nu _p^r[v](x)$$ defined variationally on balls $$B_r(x)$$ . In this paper, in a domain $$\Omega \subset \mathbb {R}^N$$ , $$N\ge 2$$ , we consider the operator $$\mu _p^\varepsilon $$ , acting on continuous functions on $$\overline{\Omega }$$ , defined by the formula $$\mu _p^\varepsilon [v](x)=\nu ^{r_\varepsilon (x)}_p[v](x)$$ , where $$r_\varepsilon (x)=\min [\varepsilon ,\mathop {\mathrm {dist}}(x,\Gamma )]$$ and $$\Gamma $$ denotes the boundary of $$\Omega $$ . We first derive various properties of $$\mu ^\varepsilon _p$$ such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function $$u^\varepsilon \in C(\overline{\Omega })$$ satisfying the Dirichlet-type problem: $$\begin{aligned} u(x)=\mu _p^\varepsilon [u](x) \ \text{ for } \text{ every } \ x\in \Omega ,\quad u=g \ \hbox { on } \ \Gamma , \end{aligned}$$ for any given function $$g\in C(\Gamma )$$ . This result holds, if we assume the existence of a suitable notion of barrier for all points in $$\Gamma $$ . That $$u^\varepsilon $$ is what we call the variational p-harmonious function with Dirichlet boundary data g, and is obtained by means of a Perron-type method based on a comparison principle. We then show that the family $$\{ u^\varepsilon \}_{\varepsilon >0}$$ gives an approximation for the viscosity solution $$u\in C(\overline{\Omega })$$ of $$\begin{aligned} \Delta _p^G u=0 \ \text{ in } \Omega , \quad u=g \ \hbox { on } \ \Gamma , \end{aligned}$$ where $$\Delta _p^G$$ is the so-called game-theoretic (or homogeneous) p-Laplace operator. In fact, we prove that $$u^\varepsilon $$ converges to u, uniformly on $$\overline{\Omega }$$ as $$\varepsilon \rightarrow 0$$ .